Thermal correlation functions in CFT and factorization

We study 2-point and 3-point functions in CFT at finite temperature for large dimension operators using holography. The 2-point function leads to a universal formula for the holographic free energy in $d$ dimensions in terms of the $c$-anomaly coefficient. By including ${\alpha }^{\mathrm{\prime }}$ corrections to the black brane background, one can reproduce the leading correction at strong coupling. In turn, 3-point functions have a very intricate structure, exhibiting a number of interesting properties. In simple cases, we find an analytic formula, which reduces to the expected expressions in different limits. When the dimensions satisfy ${\mathrm{\Delta }}_i={\mathrm{\Delta }}_j+{\mathrm{\Delta }}_k$, the thermal 3-point function satisfies a factorization property. We argue that in $d>2$ factorization is a reflection of the semiclassical regime.